Example: Finding the Domain of a Function as a Set of Ordered Pairs

Find the domain of the following function: [latex]\left\{\left(2,\text{ }10\right),\left(3,\text{ }10\right),\left(4,\text{ }20\right),\left(5,\text{ }30\right),\left(6,\text{ }40\right)\right\}[/latex] .

Answer: First identify the input values. The input value is the first coordinate in an ordered pair. There are no restrictions, as the ordered pairs are simply listed. The domain is the set of the first coordinates of the ordered pairs.

[latex]\left\{2,3,4,5,6\right\}[/latex]

Example: Finding the Domain of a Function

Find the domain of the function [latex]f\left(x\right)={x}^{2}-1[/latex].

Answer: The input value, shown by the variable [latex]x[/latex] in the equation, is squared and then the result is lowered by one. Any real number may be squared and then be lowered by one, so there are no restrictions on the domain of this function. The domain is the set of real numbers. In interval form the domain of [latex]f[/latex] is [latex]\left(-\infty ,\infty \right)[/latex].

Example: Finding the Domain of a Function Involving a Denominator (Rational Function)

Find the domain of the function [latex]f\left(x\right)=\dfrac{x+1}{2-x}[/latex].

Answer: When there is a denominator, we want to include only values of the input that do not force the denominator to be zero. So, we will set the denominator equal to 0 and solve for [latex]x[/latex].

[latex]\begin{align}2-x&=0 \\ -x&=-2 \\ x&=2 \end{align}[/latex]

Now, we will exclude 2 from the domain. The answers are all real numbers where [latex]x<2[/latex] or [latex]x>2[/latex]. We can use a symbol known as the union, [latex]\cup [/latex], to combine the two sets. In interval notation, we write the solution: [latex]\left(\mathrm{-\infty },2\right)\cup \left(2,\infty \right)[/latex]. In interval form, the domain of [latex]f[/latex] is [latex]\left(-\infty ,2\right)\cup \left(2,\infty \right)[/latex].

Example: Finding the Domain of a Function with an Even Root

Find the domain of the function [latex]f\left(x\right)=\sqrt{7-x}[/latex].

Answer: When there is an even root in the formula, we exclude any real numbers that result in a negative number in the radicand. Set the radicand greater than or equal to zero and solve for [latex]x[/latex].

[latex]\begin{align}7-x&\ge 0 \\ -x&\ge -7 \\ x&\le 7 \end{align}[/latex]

Now, we will exclude any number greater than 7 from the domain. The answers are all real numbers less than or equal to [latex]7[/latex], or [latex]\left(-\infty ,7\right][/latex].

Example: Finding the Domain and Range Using Toolkit Functions

Find the domain and range of [latex]f\left(x\right)=2{x}^{3}-x[/latex].

Answer: There are no restrictions on the domain, as any real number may be cubed and then subtracted from the result. The domain is [latex]\left(-\infty ,\infty \right)[/latex] and the range is also [latex]\left(-\infty ,\infty \right)[/latex].

Example: Finding the Domain and Range

Find the domain and range of [latex]f\left(x\right)=\dfrac{2}{x+1}[/latex].

Answer: We cannot evaluate the function at [latex]-1[/latex] because division by zero is undefined. The domain is [latex]\left(-\infty ,-1\right)\cup \left(-1,\infty \right)[/latex]. Because the function is never zero, we exclude 0 from the range. The range is [latex]\left(-\infty ,0\right)\cup \left(0,\infty \right)[/latex].

Example: Finding the Domain and Range

Find the domain and range of [latex]f\left(x\right)=2\sqrt{x+4}[/latex].

Answer: We cannot take the square root of a negative number, so the value inside the radical must be nonnegative. [latex-display]x+4\ge 0\text{ when }x\ge -4[/latex-display] The domain of [latex]f\left(x\right)[/latex] is [latex]\left[-4,\infty \right)[/latex]. We then find the range. We know that [latex]f\left(-4\right)=0[/latex], and the function value increases as [latex]x[/latex] increases without any upper limit. We conclude that the range of [latex]f[/latex] is [latex]\left[0,\infty \right)[/latex].

Analysis of the Solution

The graph below represents the function [latex]f[/latex].